NBER WORKING PAPER SERIES A THEORY OF ASSET PRICING BASED ON HETEROGENEOUS INFORMATION. Elias Albagli Christian Hellwig Aleh Tsyvinski

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1 NBER WORKING PAPER SERIES A THEORY OF ASSET PRICING BASED ON HETEROGENEOUS INFORMATION Elias Albagli Christian Hellwig Aleh Tsyvinski Working Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA October 2011 We thank Bruno Biais, John Geanakoplos, Narayana Kocherlakota, Stephen Morris, Guillaume Plantin, Jean Tirole, Martin Weber, Xavier Vives, Eric Young, and audiences at IIES Stockholm), Yale, the 2nd French Macro-Finance Summer workshop Sciences Po.), and ESSET Gerzensee for helpful comments. Hellwig gratefully acknowledges financial support from the European Research Council starting grant agreement ). Tsyvinski is grateful to NSF for support and EIEF for hospitality. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. At least one co-author has disclosed a financial relationship of potential relevance for this research. Further information is available online at NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications by Elias Albagli, Christian Hellwig, and Aleh Tsyvinski. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including notice, is given to the source.

2 A Theory of Asset Pricing Based on Heterogeneous Information Elias Albagli, Christian Hellwig, and Aleh Tsyvinski NBER Working Paper No October 2011 JEL No. E44,G12,G14,G30 ABSTRACT We propose a theory of asset prices that emphasizes heterogeneous information as the main element determining prices of different securities. Our main analytical innovation is in formulating a model of noisy information aggregation through asset prices, which is parsimonious and tractable, yet flexible in the specification of cash flow risks. We show that the noisy aggregation of heterogeneous investor beliefs drives a systematic wedge between the impact of fundamentals on an asset price, and the corresponding impact on cash flow expectations. The key intuition behind the wedge is that the identity of the marginal trader has to shift for different realization of the underlying shocks to satisfy the market-clearing condition. This identity shift amplifies the impact of price on the marginal trader's expectations. We derive tight characterization for both the conditional and the unconditional expected wedges. Our first main theorem shows how the sign of the expected wedge that is, the difference between the expected price and the dividends) depends on the shape of the dividend payoff function and on the degree of informational frictions. Our second main theorem provides conditions under which the variability of prices exceeds the variability for realized dividends. We conclude with two applications of our theory. First, we highlight how heterogeneous information can lead to systematic departures from the Modigliani-Miller theorem. Second, in a dynamic extension of our model we provide conditions under which bubbles arise. Elias Albagli University of Southern California Marshall School of Business albagli@marshall.usc.edu Christian Hellwig Toulouse School of Economics Manufacture de Tabacs, 21 Allées de Brienne, Toulouse christian.hellwig@tse-fr.eu Aleh Tsyvinski Department of Economics Yale University Box New Haven, CT and NBER a.tsyvinski@yale.edu

3 1 Introduction Dispersed investor information, and disagreement among investors about the expected cash-flows of different securities is a common feature of many, if not most financial markets. In this paper, we develop a parsimonious, flexible model of asset pricing in which heterogeneity of information and its aggregation in the market emerges as the core force determining asset prices and expected returns. Our model is tractable for a rather general specification of the asset s underlying cash-flows, and it delivers novel insights and sharp predictions that link the asset s predicted prices and returns to features of the market environment and the distribution of the underlying cash-flow risk. We further show that heterogeneous information provides a natural source of excess price volatility. Finally, our model can easily be adapted to address a variety of questions. Using our model, we reconsider two classical results in a heterogeneous information setting: the Modigliani-Miller Theorem, and the sustainability of bubbles in a dynamic environment. Specifically, we consider an asset market along the lines of Grossman and Stiglitz 1980), Hellwig 1980) and Diamond and Verrecchia 1981). 1 An investor pool is divided into informed traders who have observed a noisy signal about the value of an underlying cash flow, and uninformed noise traders. The traders all submit orders to buy shares in the cash flow at the going price. The price serves as a noisy signal of the state, which traders use along with their private signals to form an update about the cash flows. Using the market structure first introduced in Hellwig, Mukherji and Tsyvinski 2006), we assume that traders are risk neutral but face limits on their asset positions. This enables us to derive a closed-form characterization for prices and expected dividends conditional on the price, with no restriction on cash flows other than monotonicity in the underlying fundamental shocks. In our model, the asset price is equal to the expectation of cash flows for a marginal investor who is just indifferent between investing and not investing in the asset. We compare the marginal investor s posterior belief to the belief of an objective outsider, who uses the observation of the price to update beliefs about dividends, or equivalently an econometrician who uses a sample of price-dividend observations to estimate this relationship. Compared to the outsider, the marginal trader treats the information contained in the price as if he over-estimated its information content. His posterior expectations thus attach a higher weight to the market signal and a lower residual uncertainty to the fundamental than would be justified by its true information content. We label this discrepancy the information aggregation wedge. 1 See Brunnermeier 2001), Vives 2008), and Veldkamp 2011) for textbook discussions. 1

4 Despite its appearance, the information aggregation wedge is not the result of non-bayesian updating or irrational trading decisions. Instead, it results from compositional shifts under investor heterogeneity: to maintain market-clearing, the identity of the marginal trader has to change with the observed price in a way that amplifies the impact of the price on the marginal trader s expectations. For example, consider either an increase in the informed traders demand coming from a more favorable realization of the fundamentals and hence their aggregate signal distribution), or an increase in the noise traders demand. These shifts both result in a higher price and a higher expectation of future dividends, because of the information conveyed through the price. In addition, since the demand by informed traders has become larger or the pool of available securities smaller), the marginal investor s private signal has to become more optimistic just to maintain market-clearing. This further increases the price, but not expected dividends, over and above the direct signal effect. The asset price thus appears to respond more to the market signal than would be justified on the basis of its true information content. From an ex ante perspective, we characterize the average price and dividends in closed form as a function of the cash flow distribution and a parameter that summarizes the severity of the informational friction. This information friction parameter depends on the accuracy of informed traders private signals, and the variance of noise trading shocks. Intuitively, the unconditional wedge is the expected value of a mean-preserving spread of the underlying distribution of the payoffs, i.e. from an ex ante perspective the market puts a higher weight on the tails than the objective distribution. Moreover, the unconditional wedge has increasing differences between the informational noise parameter, and the asymmetry between upside and downside risks, where the latter is defined as a partial order on payoff risks that compares the marginal gains and losses at fixed distances from the prior mean of the fundamental. From this characterization, Theorem 1 then provides several general implications for expected returns. Regardless of the informational parameters, the unconditional wedge is zero when payoff risk is symmetric. The wedge is positive meaning that the expected price exceeds expected dividends) for risks that are dominated by the upside, and negative for risks that are dominated by the downside. Moreover, in absolute value this wedge becomes more pronounced for more asymmetric payoff risks, or for a higher degree of information aggregation frictions. Our model thus offers sharp, novel predictions that link the occurrence, size and direction of price premia and discounts, both unconditionally and conditionally on the realization of shocks, to specific characteristic of the market and the underlying cash-flow risk. Theorem 2 characterizes the variability of prices relative to expected and realized dividends. We 2

5 show that prices are always more variable than expected dividends. If the information aggregation wedge is sufficiently important, prices may even be more variable than realized dividends. In the limiting cases, the variability of prices exceeds that of realized dividend by any arbitrarily large factor. Moreover, the correlation between price and realized dividends may be arbitrarily close to zero. This stands in sharp contrast with the standard result in the asset pricing literature that price volatility coming from dividend expectations is bounded above by the volatility of realized dividends as in West, 1988). Since dividend volatility in the data falls short in explaining variability of prices LeRoy and Porter, 1981; Shiller, 1981), the consensus explanation stresses variation in stochastic discount rates Campbell and Shiller, 1988; Cochrane, 1992). Our theory instead suggests that high price volatility could result from volatile market expectations about dividends in a fully rational environment despite low variability in observed dividends, as long as the informational frictions are strong enough. We consider two applications of our theory. The first revisits the Modigliani-Miller Theorem, which establishes that under conditions of no arbitrage the total market value of any given cash flow is not influenced by how it is divided into separate securities. Absent distortions inside the firm, the optimal capital structure is indeterminate and disconnected from the firm s market valuation Modigliani and Miller, 1958). Capital structure theories then focus mostly on trade-offs that affect the generation of cash flows inside the firm, such as agency costs, information frictions or tax distortions, assuming that the market value of the resulting cash flow is not affected by its split into different securities. Here instead we take the view that capital structure and firm value may also be influenced by heterogeneous information and financial market frictions. We consider a seller who is splitting a given cash flow into two pieces which are sold to separate investor pools in two different markets, and suppose that at least one of the pieces is dominated either by upside or by downside risk. We show that the expected revenue of the seller is not affected by the split, if and only if the two markets are characterized by identical informational characteristics. However, when the investor pools differ, the seller can manipulate her expected revenue by selling downside risks in the market with smaller information aggregation frictions, and upside risks in the market with larger information aggregation frictions. The seller maximizes expected revenue by completely separating upside and downside risks, splitting the cash flow into a debt claim for the downside, and an equity claim for the upside, with a default point for debt at the prior median. Second, we consider the sustainability of rational bubbles. A well-known result shows that the absence of arbitrage eliminates the possibility of persistent over-pricing of securities Tirole, 1982; 3

6 Santos and Woodford, 1997). While the anticipation of higher future prices would, in principle, induce agents to increase the price bid in the current period, the combination of no arbitrage with transversality conditions or backwards induction, in case of assets with finite horizons) rules out the possibility of any security trading at a price that exceeds the net present value of expected future cash flows. We consider a simple, infinitely repeated version of our trading model with constant discounting, and give conditions under which a security may be permanently over- or under-priced, regardless of current market conditions. As usual, we can break down the current price, expected dividends and wedge into a component resulting from expectations about current cash-flows, and a component resulting from expected discounted future cash-flows and prices. The former inherits the same properties as the static conditional information aggregation wedge, while the latter inherits the properties of the unconditional wedge. If the cash-flow has a bounded downside risk and is dominated by the upside, and traders are sufficiently patient, then the positive expected future wedge more than offsets any negative current wedge. The asset then trades at a premium over its expected dividend value regardless of the current state realization. The flipside of these conditions shows that securities that have bounded upside and are dominated by the downside risk may be permanently underpriced. Finally, we discuss the theoretical robustness of the information aggregation wedge. First, we generalize distributional assumptions for fundamentals and signals, and by considering arbitrary bounds on the portfolio holdings. Second, we also extend the financial market model to include uninformed traders which partially arbitrage the wedge. Our paper contributes to a large literature on noisy information aggregation in asset markets, including the papers cited above. Much of this literature works within a canonical preference structure of CARA utility and normally distributed signals and dividends. Remarkably, the information aggregation wedge appears to have received little attention in this literature, even though it is present in these canonical models, and, as we show, is the source of rich implications for prices, trading activities, and market volatility. 2 By avoiding the restrictive functional form assumptions on cash-flow distributions, we are able to provide a characterization of this wedge for a general class of securities and draw implications that link average returns and return volatility to features of the cash-flow distribution and the importance of information frictions. Another influential literature emphasizes heterogeneous beliefs and short sales constraints as 2 The only written statement of this observation that we have found appears in Vives 2008), where it is only mentioned in passing. 4

7 potential sources of bubbles, mis-pricing, and market anomalies Harrison and Kreps, 1978; Allen, Morris and Postlewaite, 1993; Chen, Hong and Stein, 2002; Scheinkman and Xiong, 2003; Hong and Stein, 2007; Hong and Sraer, 2011). Mispricing is sustained by the option to resell an overvalued security to an even more optimistic buyer in the future. This option becomes valuable in the presence of one-sided) short-sales constraints, and implies a channel for over-valuation. Heterogeneity in prior beliefs is taken as exogenous, and with the exception of Allen, Morris and Postlewaite 1993), traders do not update from the observation of prices. We touch on similar themes, but stay within the REE tradition in which traders beliefs result from exogenous signals, and information aggregation through prices imposes tight restrictions on the heterogeneity in beliefs. Furthermore, our limits to arbitrage are not explicitly asymmetric, give rise to over- as well as under-valuation results, and our market environment is static, so the resale option doesn t play an important role except in the dynamic application to bubbles). The mechanism that gives rise to mis-pricing and bubbles is therefore quite different. 3 The literature on over-confidence explores how asset prices and financial markets are influenced by the degree to which investors over-estimate the accuracy of their own information e.g. Odean, 1998, and Daniel, Hirshleifer and Subrahmanyam, 1998). When viewed from the perspective of a representative investor, the market price that emerges in our model is perfectly consistent with these same over-confidence biases, yet all investors are fully rational and not mistaken about the quality of their signals. What may look like an over-confidence bias in the aggregate can thus be accounted for by heterogeneity and aggregation from the micro level. More generally, any theory of mispricing must rely on some source of noise affecting the market, coupled with some limits to the traders ability to exploit the resulting arbitrage opportunity see Gromb and Vayanos, 2010, for an overview and numerous references). In our model, the combination of noise trading and limits to arbitrage with heterogeneous information leads not just to random errors in the price, but to systematic, predictable departures of the price from the asset s fundamental value. The exact nature of our limits to arbitrage assumptions embedded in the position limits and the noise trading formulation) is not central for our results, but guarantees the tractability of the updating, with virtually no assumptions imposed on cash-flows. In section 2, we describe our model and provide the equilibrium characterization of asset prices. 3 For example, the difference between our work and theories of bubbles based on short-sales constraints becomes clear if one considers the case of debt instruments, as in Hong and Sraer 2011). Whereas in their model, short-sales constraints lead to over-valuation of debt, but with less volatility and trading volume than equity bubbles, our model predicts that debt may naturally be under-priced. 5

8 In section 3, we define the information aggregation wedge and discuss at length the resulting prediction for conditional and unconditional asset returns. Section 4 uses the insight offered by these two results to revisit the Modigliani-Miller theorem, and the existence of bubbles in the dynamic version of the model. Section 5 concludes the analysis with the robustness discussion. 2 Model 2.1 Agents, assets, information structure and financial market The market is set as a Bayesian trading game with a unit measure of risk-neutral, informed traders, a stochastic measure of uninformed noise traders, and a Walrasian auctioneer. There is a risky asset whose supply is normalized to a unit measure, and whose dividend is a strictly increasing and twice continuously differentiable function π ) of a stochastic fundamental θ. At the start, nature draws θ according to a normal distribution with mean 0, and unconditional variance σ 2 θ, θ N 0, σ2 θ ). Each informed trader i then receives a noisy private signal x i which is normally distributed with a mean θ and a variance β 1, and is i.i.d. across traders conditional on θ), x i N θ, β 1 ). Each trader decides whether to purchase up to one share of the asset at the prevailing price P, in exchange for cash. Formally, trader i submits a price-contingent demand schedule d i ) to maximize her expected wealth w i = d i πθ) P ). Traders cannot short-sell the asset or buy additional shares, restricting demand to [0, 1]. Individual trading strategies are then a mapping d : R 2 [0, 1] from signal-price pairs x i, P ) into asset holdings. Aggregating traders decisions leads to the aggregate demand by informed traders, D : R 2 [0, 1], Dθ, P ) = dx, P )dφ βx θ)), 1) where Φ ) denotes a cumulative standard normal distribution, and Φ βx θ)) represents the cross-sectional distribution of private signals x i conditional on the realization of θ. 4 In addition, there is stochastic demand for the asset from noise traders, which takes the form Φ u), where u is normally distributed with mean zero and variance σ 2 u, u N 0, σ 2 u), independently of θ. This specification is adapted from Hellwig, Mukherji, and Tsyvinski 2006), and allows us to preserve the tractability of Bayesian updating with normal posterior beliefs. 5 Once all traders have submitted their orders, the auctioneer selects a price P to clear the market. Formally, the market-clearing price function P : R 2 R selects P from the correspondence 4 We assume that the Law of Large Numbers applies to the continuum of traders, so that conditional on θ the cross-sectional distribution of signal realizations ex post is the same as the ex ante distribution of traders signals. 5 We generalize this demand specification in Section 4.2 allowing for price-elastic demands by noise traders. 6

9 ˆP θ, u) = {P R : Dθ, P ) + Φu) = 1}, for all θ, u) R 2. 6 After all trades are completed, the dividends πθ) are realized and disbursed to the owners of the asset. Let H x, P ) : R [0, 1] denote the traders posterior cdf of θ, conditional on observing a private signal x, and conditional on the market price P. A Perfect Bayesian Equilibrium consists of demand functions dx, P ) for informed traders, a price function P θ, u), and posterior beliefs H x, P ) such that i) dx, P ) is optimal given H x, P ); ii) the asset market clears for all θ, u); and iii) H x, P ) satisfies Bayes rule whenever applicable, i.e., for all p such that {θ, u) : P θ, u) = p} is non-empty. 2.2 Equilibrium Characterization We begin by characterizing informed traders demand. With risk-neutrality, the trader s expected value of holding the asset is πθ)dh θ x, P ). Since private signals are log-concave and π ) is increasing in θ, posterior beliefs H x, P ) are first-order stochastically increasing in x, and πθ)dh θ x, P ) is strictly increasing in x, for any P that is observed in equilibrium Milgrom, 1981). The traders decisions are therefore characterized by a signal threshold function ˆx : R R {± }, such that dx i, P ) = I xi ˆxP ), that is, the trader places an order to buy a share at price P, if and only if x i ˆxP ). We call the trader who observes the signal equal to the threshold, x = ˆx P ), and who is therefore indifferent, the marginal trader. The signal threshold is uniquely defined by ˆxP ) = + if lim πθ)dh θ x, P ) P, x + ˆxP ) = if lim πθ)dh θ x, P ) P, x P = πθ)dh θ ˆxP ), P ) otherwise. 2) Expression 2) illustrates three cases: i) if the most optimistic trader s expected dividend is lower than the price, no trader buys, so the signal threshold becomes + ; ii) if the most pessimistic trader s expected dividend exceeds the price, all traders buy, and the threshold for buying is ; iii) only some traders buy, and the threshold ˆxP ) takes an interior value at which the marginal trader s posterior expectation of the dividend must equal the price. Aggregating the individual trading decisions, the informed demand is Dθ, P ) = ˆxP ) 1 dφ β x θ)) = 1 Φ β ˆxP ) θ)), which equals 0 if ˆxP ) = +, and 1 if ˆxP ) =. 6 We can without loss of generality restrict the range of P ) to coincide with the range of π ). 7

10 Next, we analyze the market-clearing condition. Since Φu) 0, 1), in equilibrium, ˆx ) must be finite for all P on the equilibrium path, and satisfy the third condition in 2). From the marketclearing condition, we then have Φ β ˆxP ) θ)) = Φ u), which allows us to characterize the correspondence of market-clearing prices: { ˆP θ, u) = P R : ˆxP ) = θ + 1 } u. 3) β From now on, we focus on equilibria in which the price is conditioned on θ, u) through the observable state variable z θ + 1/ β u. The next lemma characterizes the resulting equilibrium beliefs. All proofs are provided in the appendix. Lemma 1 Information Aggregation) i) In any equilibrium with conditioning on z, the equilibrium price function P z) is invertible. along the equilibrium path are given by H θ x, P ) = Φ σ 2 θ ii) Equilibrium beliefs for price realizations observed + β + βσ 2 u )) βx + βσ 2 u ˆxP ) θ + β + βσu 2. 4) Part ii) of the Lemma exploits the invertibility to arrive at a complete characterization of posterior beliefs H x, P ). With invertibility, we can summarize information conveyed by the price through z. Conditional on θ, z is normally distributed with mean θ and variance σ 2 u/β. Thus, the price is isomorphic to a normally distributed signal of θ, with a precision that is increasing in the precision of private signals, and decreasing in the variance of demand shocks. Using Lemma 1 we rewrite 2), the indifference condition that defines the signal threshold ˆxP ): )) P = πθ)dφ σ 2 θ + β + βσu 2 β + βσu 2 θ ˆxP ). 5) σ 2 θ σ 2 θ + β + βσ 2 u This condition equates P to the marginal trader s expectation of dividends. The latter also depends on P through its effect on posterior beliefs. Using the market-clearing condition ˆxP ) = z, Proposition 1 uniquely characterizes the market equilibrium. 7 Proposition 1 Asset market equilibrium) For any increasing dividend function π ), an asset market equilibrium exists, is unique, and is characterized by the price function P π z) and the traders threshold function ˆxp) = z = Pπ 1 p), where P π z) = E πθ) x = z, z) = πθ)dφ σ 2 θ + β + βσu 2 θ σ 2 θ β + βσ 2 u + β + βσ 2 u z )). 6) 7 Notice that this only implies the uniqueness of the equilibrium that conditions on the summary statistic z, not overall uniqueness of the equilibrium characterized in proposition 1. 8

11 The price function P π z) is uniquely defined and strictly monotone, and therefore defines the unique market equilibrium. 8 3 The Information Aggregation Wedge 3.1 Conditional Information aggregation wedge We now discuss how noisy information affects equilibrium prices and expected dividend values. To be precise, we form expectations of dividends from the perspective of an outside observer or econometrician ) who does not have access to any private signal about θ, but knows the parameters of the game and observes the realization of the price P, or equivalently the state z. This outsider holds a conditional belief that θ z N βσu 2 / σ 2 θ + βσu 2 ) z, σ 2 θ + βσu 2 ) 1 ), and therefore has an expectation of dividends conditional on public information z, denoted V π z): )) V π z) = E πθ) z) = πθ)dφ σ 2 θ + βσu 2 βσu 2 θ z. 7) σ 2 θ + βσ 2 u The main observation from comparing Proposition 1 with equation 7) is that at the interim stage when the share price is observed but before dividends are realized the equilibrium price differs from the expected dividend, conditional on the public information. This difference is due to the impact of private information on equilibrium prices. We label this difference the information aggregation wedge, W π z) P π z) V π z). The choice of V π ) as a natural benchmark of comparison for P π ) follows from the fact that V π ) also corresponds to the expected dividend or in a sufficiently large data set, to the average dividend), conditional on the observation of P recall that P π ) is invertible). This benchmark differs from the one chosen e.g. by Harrison and Kreps 1978), who compare an asset s value to the dividend expectation of any trader in their market, or to an average of those expectations, as in Bacchetta and van Wincoop 2006). Our formulation has the advantage that V π ) and P π ) both have direct empirical counterparts in any set of price-return data, and this formulation therefore allows us to directly focus on the empirical, testable implications of our model. The price P π z), and the expected dividend conditional on public information, V π z), differ in how expectations of θ are formed. The price equals the dividend expectation of the marginal trader who is indifferent between keeping or selling her share. This trader conditions on the market signal z, as well as a private signal whose realization must equal the threshold ˆxP ) in order to be 8 We index an equilibrium function or variable by π to make explicit that it is derived from a specific dividend function π ), i.e. P π ) is the equilibrium price function that is derived from dividend function π ) by equation 6). 9

12 consistent with the trader s indifference condition. The trader treats these two sources of information as mutually independent signals of θ. At the same time, the market-clearing condition implies that ˆxP ) must equal z in order to equate demand and supply of shares. The marginal trader s expectation E πθ) x = z, z) thus behaves as if she received one signal z of precision β + βσ 2 u instead of βσu 2. In contrast, the expected dividends E πθ) z) conditional on P or equivalently z) weighs z according to its true precision βσ 2 u. Figure 1: Marginal Trader Identity Shift The difference in the responsiveness of the price relative to the expected dividend conditional in the price results from the compositional shift in the identity of the traders holding the shares. This is depicted in figure 1. Any increase in z shifts the identity of the marginal trader s private signal one-for-one. If θ increases, the distribution of private signals shifts up, so for a given signal threshold, demand for the asset by informed traders increases, but demand from uninformed traders is unchanged. If instead s increases, uninformed demand increases, but informed demand remains the same. In both cases, the asset is relatively scarcer for informed traders, so the threshold for the informed trader s private signal has to increase in order to clear the market. In addition to this compositional shift which only appears in the expectation of the marginal trader, e.g. the price), all traders, as well as the uninformed outsider, recognize that an increase in z, as revealed through P shifts up their expectation of the state θ. This is reflected in the weight βσ 2 u both P π z) and V π z). attributed to z in Belief heterogeneity and limits to arbitrage are both necessary ingredients to obtain the wedge. 10

13 If instead all informed traders have access to a common signal z of fundamentals, they all hold identical expectations and must be indifferent between buying and not buying to clear the market. But this requires that the price equals the common expectation of the dividend, i.e. P π z) = V π z). The same result applies with free entry of uninformed arbitrageurs Kyle, 1985). The remainder of this subsection describes properties of the wedge, conditional on z, which will form the basis for our main results on expected returns and price volatility. To this end, we define γ P σ 2 θ β + βσ 2 u + β + βσ 2 u, and γ V σ 2 θ βσ 2 u + βσ 2 u as the response coefficients of the expectations of θ entering P π ) and V π ) to innovations in z. The equilibrium price and expected dividends are then rewritten as: P π z) = πγ P z + σ θ 1 γp u)φ u) du V π z) = πγ V z + σ θ 1 γv u)φ u) du This formulation summarizes the difference between the price and the expected dividend by the response parameters γ P and γ V, which measure the marginal trader s and outsider s update of θ to z. These parameter enter P π ) and V π ) in two ways: the marginal trader s expectation responds more strongly to z, and his residual uncertainty about θ after observing z) is lower: σθ 2 1 γ P ) instead of σθ 2 1 γ V ). Using a third-order Taylor expansion, we approximate the wedge by W π z) π γ P z) π γ V z) + σ2 θ 2 [ π γ P z) 1 γ P ) π γ V z) 1 γ V ) ]. 8) The term π γ P z) π γ V z) captures the shift in expectations, while the second term in squared brackets captures the role of residual uncertainty. The latter plays a role only if π ) is non-linear, and in that case matters through second- and higher derivatives. The shift in expectations from γ V z to γ P z amounts to a mean-preserving spread from an ex ante perspective, and is therefore a source of increased variability in the price, relative to expected fundamentals: π γ P z) π γ V z) crosses 0 at a single point where z = 0, is negative when z < 0, and positive when z > 0. When π ) is linear, the higher sensitivity of expectations to z is the only effect determining the wedge, while the residual uncertainty plays no role. Panel a) of figure 2 plots the price solid line), the expected dividend dashed line) and conditional wedge dashed-dotted line) as a function of the state variable z, for πθ) = θ. The price is more sensitive to innovations in z than the expected dividend, resulting in a wedge W π z) = γ P γ V )z that is negative for z < 0, zero for z = 0, and positive for z > 0. 11

14 For non-linear dividends, residual uncertainty shifts the level of the wedge up or down, depending on a comparison between the residual uncertainty levels 1 γ P relative to 1 γ V, and the second derivatives π γ P z) and π γ V z). At the prior mean z = 0, the second derivatives are comparable, so the reduction of uncertainty implies a negative wedge if π 0) > 0, and a positive wedge, if π 0) < 0. Away from z = 0, the third- and higher derivatives may reduce or even overturn this effect, and therefore make it impossible to offer precise results on the shape of W π ) without additional restrictions. We illustrate these possibilities with two parametric examples that follow. Figure 2: Conditional Price, Expected Dividend, and Wedge a) Linear dividend function Pz) Vz) Wz) b) Cubic dividend function a = 10) Pz) Vz) Wz) > z st. dev.) > z st. dev.) c) Exponential dividend function k = 2) Pz) Vz) Wz) 10 0 d) Exponential dividend function k = 2) Pz) Vz) Wz) > z st. dev.) > z st. dev.) Example 1: Exponential dividend function Suppose that π θ) = 1 k ekθ, with k 0. Expected dividends, prices and the wedge are then 12

15 characterized by: V π z) = 1 k ekγ V z+ k2 2 σ2 θ 1 γ V ), P π z) = 1 k ekγ P z+ k2 2 σ2 θ 1 γ P ) ) W π z) = P π z) 1 e kγ P γ V )z+ k2 2 γ P γ V )σθ 2. In this case, the price and expected dividend are both exponential functions in z, with a stronger reaction of prices to z. The residual uncertainty affects both V π z) and P π z) multiplicatively, but the factor is larger for V π z), reflecting the fact that residual uncertainty is greater for the outsider. If k > 0, we then have a dividend function that is increasing, convex, and bounded below by zero figure 2, panel c). The wedge is negative at z = 0 and non-monotone. It decreases at first, reaches its lowest value at some intermediate point, and is increasing and convex from there on, crossing 0 at z = k 2 σ2 θ > 0. The reverse image obtains when k < 0, in which case π is increasing, concave, and bounded above by zero figure 2, panel d). For negative z, the wedge is negative at first and increasing in z, crossing 0 at z = k 2 σ 2 θ < 0. It reaches its maximum value at a negative z and then monotonically converges towards 0. This example thus confirms the intuitions from the shift in means which makes P π z) more responsive to a shift in z, and the shift in residual uncertainty that is captured by the multiplicative factors. The curvature parameter k governs the shape of the wedge function, and whether the residual uncertainty increases or decreases the wedge. We use this example to illustrate our two main results. First, we show that the expected wedge is positive if and only if k > 0, and negative if k < 0. That is, the security trades at a premium in the case with convex dividends and upside risks, and at a discount in the case with concave dividends and downside risks. Taking expectations, we have [ E V π z)) = 1/k e k2 2 σ2 θ, E Pπ z)) = 1/k e k 2 γp 2 σ2 θ 1+ { } and E W π z)) = 1/k e k2 2 σ2 θ e k2 2 σ2 θ γ P /γ V 1)γ P 1, ] γv 1 )γ P which is positive whenever k > 0, and negative for k < 0 and can be checked to approach 0 continuously as k 0). Second, we show that the model exhibits excess price volatility. for analytical convenience, we have: Focusing on log variances V ar log π θ)) = k 2 σ 2 θ, V ar log V π z)) = γ V k 2 σ 2 θ, and V ar log P π z)) = γ 2 P /γ V k 2 σ 2 θ. Therefore, we observe that V ar log P π z)) > V ar log V π z)), for any parameter set. Moreover, if the information aggregation wedge becomes sufficiently important, then we may have γ 2 P /γ V > 1, and therefore V ar log P π z)) > V ar log π θ)). In particular, this is a result of the following two limiting scenarios: i) if for given γ V < 1, γ P approaches 1, i.e. the informed traders have very precise signals for given level of information in the price, or ii) if γ V 0, 13

16 while γ P is bounded away from 0. In this case, the market becomes very noisy, for a given level of private information. On the other hand, V ar log V π z)) is always less than V ar log π θ)), which is a direct application of Blackwell s Theorem on comparison of information structures Blackwell, 1951, 1953). Our main two theorems that follow generalize these observations about the unconditional wedge and excess price volatility. Theorem 1 below establishes that the sign and magnitude of the average wedge on the comparison of upside vs. downside risks. Theorem 2 generalizes the result that prices are more variable than expected dividends, and in some cases even more variable than realized dividends. Our second example, however, reinforces the observation that the conditional wedge W π ) need not be monotone in general, and may also cross 0 at multiple points, which rules out conditional or local versions of these results without imposing additional assumption on dividends. Example 2: Cubic dividend function Suppose that π θ) = θ + aθ 3, with a > 0 to ensure monotonicity of π. For a cubic function figure 2, panel b), the above approximation holds exactly, so that W π z) = γ P γ V )z + aγp 3 γv 3 )z 3 + 3azσθ 2 [γ P 1 γ P ) γ V 1 γ V )], where the first two terms correspond to the shift in means, and the last to the shift in residual uncertainty. If γ P + γ V > 1 and a sufficiently large, W π0) < 0. Since W π 0) = 0, it follows immediately that W π ) is non-monotone and crosses 0 in three different locations. 3.2 Unconditional information aggregation wedge To obtain general results, we focus on unconditional moments of prices and expected dividends. Let W π = E W π z)) denote the expected information aggregation wedge associated with a payoff function π ). The next lemma provides a characterization of W π which forms the basis for the subsequent comparative statics results. Lemma 2 Unconditional Wedge) Define σ P as σ 2 P = σ2 θ 1 + γ P /γ V 1) γ P ). The unconditional information aggregation wedge W π is characterized by W π = 0 π θ) π θ) ) ) )) θ θ Φ Φ dθ. 9) σ θ σ P This characterization shows how the wedge depends separately on both the curvature the payoff function, and the parameters describing the informational environment. The parameter σ P > σ θ 14

17 corresponds to the prior variance of θ, as assessed by the marginal trader, and summarizes the importance of informational frictions in the market. By taking ex ante expectations over z, the shifts in mean and residual uncertainty combine into a mean-preserving spread between the weights that the marginal trader and the outsider associate with each realizations of θ. The marginal trader places more weight on the tails of the fundamental distribution, from an ex ante perspective i.e., σ P > σ θ ). This result can intuitively be understood as follows: the marginal trader s posterior of θ, conditional on z, is normal with mean γ P z and variance 1 γ P ) σ 2 θ. The prior over z is normal with mean 0 and variance σ 2 θ /γ V. Compounding the two distributions, the marginal trader s prior over θ is characterized as a normal distribution with mean 0 and variance 1 γ P ) σθ 2 + γ2 P σ2 θ /γ V = σp 2. The outsider, on the other hand, holds the posterior that conditional on z, θ is normal with mean γ V z and variance 1 γ V ) σ 2 θ. His compounded distribution then corresponds to the actual prior distribution of θ, as the prior variance is just 1 γ V ) σ 2 θ + γ V σ 2 θ = σ 2 θ. Hence, the information frictions summarized by the distance of σ P from σ θ will be large whenever the market signal is noisy relative to private signals, or the ratio γ P /γ V is high, as this leads to a large discrepancy between the posterior beliefs held by the marginal trader and the outsider. We use Lemma 2 to sign the unconditional wedge as a function of the shape of the dividend function, and to offer comparative statics with respect to π and the informational parameters γ P and γ V. Our next definition provides a partial order on payoff functions that we will use for the comparative statics. Definition 1 i) A dividend function π has symmetric risks if π θ) = π θ) for all θ > 0. ii) A payoff function π is dominated by upside risks, if π θ) π θ) for all θ > 0. A payoff function π is dominated by downside risks, if π θ) π θ) for all θ > 0. iii) A dividend function π 1 has more upside less downside) risk than π 2 if π 1 θ) π 1 θ) π 2 θ) π 2 θ) for all θ > 0. This definition classifies payoff functions by comparing marginal gains and losses at fixed distances from the prior mean to determine whether the payoff exposes its owner to bigger payoff fluctuations on the upside or the downside. Any linear dividend function has symmetric risks, any convex function is dominated by upside risks, and any concave dividend function is dominated by downside risks. The classification however also extends to non-linear functions with symmetric gains and losses, as well as non-convex functions with upside risk or non-concave functions with downside risk. Figure 3 plots examples of payoff functions dominated by different types of risk. 15

18 Figure 3: Dividend risk types π θ ) upside risk symmetric risk downside risk > θ st.dev.) The following Theorem summarizes the comparative statics implications that follow directly from this partial order, and the characterization in lemma 2. Theorem 1 Average prices and returns) i) Sign: If π has symmetric risk, then W π = 0. If π is dominated by upside risk, then W π 0. If π is dominated by downside risk, then W π 0. ii) Comparative Statics w.r.t. π: For given σ 2 P, if π 1 has more downside and less upside risk than π 2, then W π2 W π1. iii) Comparative Statics w.r.t. σp 2 : If π is dominated by upside or downside risk, then W π is increasing in σ P. Moreover, lim σp σ θ W π = 0, and lim σp W π =, whenever there exists ε > 0, such that π θ) π θ) > ε for all θ ε. iv) Increasing differences: If π 1 has more upside risk than π 2, then W π1 σ P ) W π2 σ P ) is increasing in σ P. This theorem summarizes how the shape of the dividend function and the informational parameters combine to determine the sign and magnitude of the unconditional information aggregation wedge. It shows that unconditional price premia or discounts arise as a combination of two elements: upside or downside risks in the dividend profile π, and an impact of private information on market prices γ P > γ V ). The latter requires that updating from prices is noisy γ V < 1). This Theorem forms the first part of our core theoretical contribution, and shows that noisy information aggregation may influence conditional and unconditional returns of assets through their payoff 16

19 profile and the informational characteristics of the market. The result is easily understood from our interpretation of the wedge as the expected value of a symmetric, mean-preserving spread of the true underlying fundamental distribution. Part i) shows that the sign of the wedge is determined by whether π is dominated by upside, downside, or symmetric risk. When the dividend function has symmetric risk, the gains from this spread on the upside exactly cancel the expected losses on the downside, and the total effect is 0. When the dividend is dominated by upside risks, the expected upside gains dominate and the value of the mean-preserving spread is positive, leading to a positive unconditional wedge. Conversely, when the dividend is dominated by downside risks, the expected losses on the downside dominate and the expected value of the spread is negative. Parts ii), iii), and iv) complement the first result on the possibility of price premia or discounts with specific predictions on how its magnitude depends on cash flow and informational characteristics. Part ii) shows that an asset with more upside or less downside risk on average has a higher price premium or a lower price discount, all else equal. Thus, returns on average are lower and prices higher) for securities that represent more upside risks. Simply put, the mean-preserving spread becomes more valuable when the payoff function shifts towards more upside risk. Part iii) shows the role of informational parameters. For a given payoff function, the unconditional wedge increases in absolute value as the information aggregation friction has bigger effects higher σ P ). For a given set of upside or downside risks, a bigger mean-preserving spread generates bigger gains or losses. Moreover, a wedge obtains only if γ P > γ V, i.e., if the heterogeneous beliefs have an impact on price. The wedge is increasing in γ P and decreasing in γ V, as the precision of market information and private information move the wedge in opposite directions. Under regularity conditions, which ensure that the payoff asymmetry doesn t disappear in the tails, the absolute value of the wedge approaches infinity when γ V 0. This obtains if for a given value of β, the market noise becomes infinitely large. In this limiting case, the marginal trader remains responsive to z, even though the z is infinitely noisy. Part iv) shows that the unconditional wedge has increasing differences between the dominance of upside risk and the level of market noise. This implies that the effects of market noise and asymmetry in dividend risk are mutually reinforcing on the magnitude of the wedge. Importantly, our results on differences between expected prices and dividends are not a consequence of irrational trading strategies, behavioral biases of investors, or agency conflicts. Nor are such differences accounted for by risk premia since traders are risk neutral). Our model thus offers 17

20 a theory in which average prices can differ systematically from expected dividends as a result of the interplay between the dividend structure and the partial aggregation of information into prices, in a context where traders hold heterogeneous beliefs in equilibrium and arbitrage is limited. To our knowledge, this channel is new to the literature. 3.3 Excess Price Variability Our second main result concerns the variability of prices, relative to expected dividends and realized dividends. As can readily be seen from the above characterizations, if W π ) > 0, the unconditional variance of prices prior to realization of z) exceeds the variability of expected dividends. Consider furthermore the limiting case where γ P 1, in which P π z) π z). Since the variance of z exceeds that of θ, it follows immediately that in this limit, where the informed traders signals become arbitrarily precise, the variability of prices can exceed the variability of dividends. This result is illustrated in the linear and the log-normal examples discussed in section 3.1. Our second theorem generalizes these observations. To do so, we will need to impose some restrictions to handle the non-linearities and higher-order effects that are confounding the comparative statics of W π with respect to z. Concretely, we will focus on risks that are symmetric or dominated by the upside or downside, and we will focus on E P π z) P π 0)) 2), E V π z) V π 0)) 2), and E π θ) π 0)) 2) as our criterion for the variability of prices, expected dividends, and realized dividends, respectively, rather than the unconditional variances. The next theorem states our main result concerning excess variability: Theorem 2 Excess variability of prices) For any payoff function π ) that is symmetric, dominated by upside, or dominated by downside risk: i) The variability of expected dividends is always less than the variability of realized dividends and the variability of prices: E V π z) V π 0)) 2) < E π θ) π 0)) 2) and E V π z) V π 0)) 2) < E P π z) P π 0)) 2) ii) The excess variability of prices relative to expected dividends is increasing in γ P and decreasing in γ V. iii) For any γ V, if γ P is sufficiently high, then the variability of prices exceeds the variability of realized dividends. The same occurs if, for given γ P, γ V is sufficiently low. iv) If π ) is unbounded on one side, then lim γv 0 E P π z) P π 0)) 2) =. 18

21 This theorem shows that the price is more variable than expected dividends, and if the market is sufficiently noisy, even more variable than realized dividends. The latter occurs in the limiting cases where supply shocks are unboundedly large σ 2 u, γ V information is infinitely precise β, γ P 0), or the traders private 1). In the former case, the variability of prices can be arbitrarily large, even as the variability of realized and expected dividends is bounded. The statement of the result relies on two restrictions which we used for analytical tractability. First, the focus on a variability measure which combines a variance with a bias between the average price and the price at the average fundamental. Second, we restrict ourselves to symmetric, upside or downside risks. With these restrictions, the results are the cleanest, and easiest to interpret. To understand this result, and the source of excess price variability in our model, it is useful to think of a counter-factual third person who observes a signal z with distribution z θ N θ, β + βσu 2 ) 1 ). Like the uninformed outsider, this third person is fully Bayesian, but has access to a more informative signal, whose precision matches that of the marginal traders. Therefore in comparison to the marginal trader, the third person will form the same posterior beliefs, conditonal on a realization of z, but z will be drawn from a distribution with a lower ex ante variance, and be consistent with Bayes Rule derived from the objective signal precision. In comparison to the uninformed outsider, the third person is also fully Bayesian, but with simply a more precise signal. We break down the comparison between E P π z) P π 0)) 2) and the other terms into a comparison between E P π z) P π 0)) 2) ) and E P π z) P π 0)) 2 z N θ, β + βσu 2 ) 1 ), and the comparison of this latter term with the ex ante variability of expected and realized dividends. E P π z) P π 0)) 2 z N θ, β + βσu 2 ) 1 corresponds to the counter-factual ) variability of prices, if z had been drawn from a distribution z θ N θ, β + βσu 2 ) 1 ), such that P π z) is consistent with a posterior expectation of π conditional on z. For the comparison of the counter-factual variability of prices with the variability in expected and realized dividends, we first proceed to break down the variability measures into a variance and a bias term. The variance terms can then be compared using Blackwell s theorem on the comparison of experiments Blackwell, 1951, 1953). Since π θ), P π z), and V π z) correspond to the posterior expectation of π θ) for respectively, an agent who observes the true θ, the counter-factual signal z, and the actual signal z, the unconditional variance of π exceeds the unconditional variance of P under the distribution z N θ, β + βσu 2 ) 1 ), which exceeds the unconditional variance of V under the distribution z N θ, σ 2 u/β). For symmetric, upside and downside risks, the bias terms follow exactly the same ranking. 9 9 Our choice of variability measure which is equivalent to the variance for symmetric risks) allows for the cleanest 19